Reconstruction of metric measure spaces and the spectral information on distances

The famous Gromov-Vershik metric measure space reconstruction theorem gives a way to reconstruct uniquely the metric measure space (up to a measure preserving isometry) by a suitable information on distances between randomly chosen points in the latter. The natural questions are then what can be reconstructed from the spectral information on random distance matrices. We will discuss some of these problems and their relationship to the multidimensional scaling, a widely used method in manifold learning.